#350027

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25605	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(11,19)(12,13)(15,22)(16,17)(20,21)(23,24)(27,33)(28,35)(30,37)(31,39)(34,36)(38,40);(2,5)(6,8)(9,10,14,18)(11,15,20,23)(12,16,13,17)(19,22,21,24)(25,26,29,32)(27,30,34,38)(28,31,35,39)(33,37,36,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,22)(18,23)(25,27,29,34)(26,30,32,38)(28,36,35,33)(31,40,39,37);(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,24)(18,23)(28,35)(31,39)(33,36)(37,40)

Polycyclic presentation:

2^9|b^2=h,c^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257223);